Peter Sprent

Three or more samples

For parametric methods based on normal distribution theory the extension from one or two to several samples shifts emphasis from t-tests and the t-distribution to the analysis of variance and the F-distribution (related because t2 with v degrees of freedom has an F-distribution with 1, v degrees of freedom). Wilcoxon type tests are nonparametric analogues of t-based procedures for the one- or two-sample case. Extending to three or more samples, we find relevant nonparametric techniques closely parallel analysis of variance methods — particularly in computational aspects. The test procedures, especially for moderately large samples, will involve normal approximations and are often based on the F-distribution or the chi-squared distribution (see Section A5).

Tests and estimation for two independent samples

When we have two independent samples and wish to compare their location (medians or means) we can no longer reduce this to a one-sample problem.

Methods for paired samples

Single samples are useful for illustrating basic ideas, but cover only limited applications. More usually, we are faced with (i) two or more independent samples, or (ii) two or more related samples. In this and the next chapter we limit discussion to two samples, dealing in this one with paired samples, a special class of related samples for which many problems reduce to a single-sample equivalent.

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Covers all areas of the mathematics of business and finance including: quality control, production scheduling, optimization, inventory control, stocking policy and simulation planning.

Bivariate and multivariate data

There are several nonparametric methods for correlation and regression. We consider first the bivariate case where methods are generally applied to a sample of bivariate observations (x i , y i ), i = 1, 2,..., n that are at least ordinal. If they are measurements or counts we sometimes convert these to ranks and, in the case of correlation, base our statistics on these ranks. Depending on the precise problem, procedures for regression will sometimes use ranks and sometimes the original observations.

Robustness, jackknives and bootstraps

In this chapter we adopt a different approach and review several important techniques with applications ranging from simple location estimates and tests to complicated regression problems. All require appropriate computer programs for realistic applications and a general understanding of fairly advanced statistical theory to make best use of them and to avoid some of the pitfalls that go with statistical sophistication (indeed with sophistication in most applied science). We can, however, indicate the principles by numerically trivial examples that demonstrate the rationale of each approach. The experimenter who is not a statistical expert should be aware these methods exist; he or she may then seek a statistician’s advice about their application to particular problems.

Introducing nonparametric methods

In most of this book we assume only a rudimentary knowledge of statistics like that provided by a service or introductory course of 10 to 20 lectures, or by reading a simple text like Statistics Without Tears (Rowntree, 1981).

Location estimates for single samples

In Chapters 2 to 7 we consider data that are either measurements or ranks specifying order of magnitude or preference. The latter are called ordinal data. Chapters 2 and 3 are devoted to single samples. Most practical problems involve comparison of, or studying relations between, several samples, but many basic nonparametric notions are applicable also to a single sample and the logic behind them is easily explained in this basic situation.

Counts and categories

Data often consist of counts of numbers of objects with given attributes or belonging to given categories arranged into one-, two-, three- or even higher-dimensional tables, usually referred to as one-, two-, three-way contingency tables. Each dimension or ‘way’ corresponds to a classification into categories representing one attribute.

Distribution tests and rank transformations for single samples

Chapter 2 covered test and estimation procedures for location — the median if we do not assume symmetry; either mean or median if we do. We now consider more general problems of matching data to particular distributions. Populations, whether or not they have identical locations, may differ widely in other characteristics.